In Exercises 31 to 48 , find f^-1(x). State any restrictions on the domain of f^-1(x). f(x)=x^2+

In Exercises 31 to 48 , find f^-1(x). State any restrictions...

Key Concepts

Quadratic Functions

Quadratic functions, typically written in the form f(x) = ax² + bx + c, produce parabolic graphs. Since quadratic functions are not one-to-one over their natural domain, their domain is often restricted to ensure that the function passes the horizontal line test. This restriction is necessary when finding an inverse, as it guarantees that the function is invertible.

Completing the Square

Completing the square is an algebraic method used to rewrite a quadratic function in vertex form. This technique facilitates the process of finding the inverse by making it easier to solve for x in terms of y, particularly when isolating the squared term. It also provides insight into the function’s transformation and vertex, which aids in understanding the function's behavior and appropriate domain restrictions.

Inverse Function

An inverse function reverses the operation of the original function by mapping outputs back to their corresponding inputs. For a function f and its inverse f?¹, the relationship f(f?¹(x)) = x and f?¹(f(x)) = x holds true. In order for an inverse to exist, the original function must be bijective, meaning it must be one-to-one and onto on the domain considered.

Domain and Range Restrictions

When finding an inverse function, the domain and range switch roles. It is essential to restrict the original function’s domain so that it is one-to-one, ensuring that every output corresponds to a unique input. The range of the original function then becomes the domain of the inverse function, and any limitations on the range must be carried over as restrictions on the inverse.

Transcript

00:01 So the first thing we're being asked to do in this problem is to find the inverse of x.

00:05 That's what f negative 1 of x means.

00:07 Means to find the inverse of a function, it's found by switching our x and y values.

00:13 Remember, f of x is the same thing as y.

00:16 So the first thing we're going to do is in place of y, we're going to put in x.

00:21 And in place of both of our xes, we're going to put in y.

00:24 So have y squared plus four y.

00:27 So now what we need to do is isolate by.

00:30 Essentially, we got to solve this equation for y.

00:32 Okay.

00:33 Well, even if we factor out of y, we're going to still have a separate y term.

00:37 So what we're going to do is do completing the square on the right hand side.

00:41 So in other words, i'm trying to find the constant that will make this a perfect where trinomial.

00:46 To do this, we use the formula, b over 2 squared, where b is the coefficient of our y term, which is 4.

00:52 So we'll have 4 divided by 2, which is 2, and 2 squared is 4.

00:56 So i'm going to add 4 to the right hand side.

00:59 But if i add four to the right, we better add four to the left.

01:03 So on the left -hand side, we'll have x plus four...

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